Here, we use the short-hand notation \(k=({\bf k},\sigma )\)  to specify both the
wavevector and the phonon branch, and \(-k\)  means \((-{\bf k},-\sigma )\) . In normal lattice dynamic treatment,
we usually take \(\sigma , \omega \ge 0 \)  as a convention, and require
\(\omega _{\sigma ,{\bf k}}=\omega _{\sigma ,-{\bf k}}\) . For the current problem,
this is not true [1]}, [2]}. It is more convenient
to have the frequency taking both positive and negative values and require
the above equation (REF ).
And from Eq. (),
the momentum and displacement polarization vectors are related through
\(\mu _k = -i\omega _k \epsilon _k + A\epsilon _k.\) 
